There seems to be no perfect definition: sources tend to give descriptions or examples. It does not depend on denomination, though different denominations might focus on different things.
The tension is, roughly, between the facts that:
- Christ came and redeemed us by His Sacrifice; now we are Christians, living in this present world, with duties to love God and others
- Christ has not yet come again, all has not yet come to pass; we are not yet at the end, though as Christians we strive towards it
This tension is not necessarily a bad thing. If there is concern, it is perhaps with directing ourselves properly. For example, a person might focus too much on yearning for heaven, feeling too strongly its absence in this world, and fall into neglecting or even despising worldly Christian duties and life. Or an individual might commit too much to this world, perhaps trying to solve its problems, or feeling too crushed by its problems, and forget where we are all going. Or an individual might feel no tension at all, no pull towards either good thing -- or too much of a pull from both directions. These are just examples: they're a bit at the extreme ends of the spectrum, and might be off the mark in certain contexts.
(end - the rest of this answer gives examples of usage in articles centered on various denominations)
From The Modern Eschatological Debate (pdf), in the Evangelical Quarterly:
That is just the way in which it is given in Calvin, for with him
every doctrine entails what we now call the eschatological
[...] That means that the eschatological tension that reached its acutest point in the person and work of the Mediator is enshrined in the tension between justus et peccator and between real presence and bread and wine. To the word of justification the sacraments add the pressure of imminence so that the two together on their Christological basis contain the heart of eschatology.
From an article in the Journal of Lutheran Ethics:
In Romans 5 Paul writes of the two aeons: the new era that Christ is bringing into the world and the old aeon that is under the rule of law and sin. The same eschatological tension is present in other biblical sources. The new order of Christ is in tension with the old order, yet Christians must live in both. [...]
All major Christian religious traditions recognize in some fashion this tension between Christ and the ongoing societal necessities of the world. They are aware that following Christ and living in the world is no easy task.
(The above article makes comments that some might find misrepresent other traditions, or are biased towards the Lutheran view.)
Pope John Paul II discusses the interplay between this tension and the Eucharist. From Ecclesia de Eucharistia, Church of the Eucharist:
The eschatological tension kindled by the Eucharist expresses and reinforces our communion with the Church in heaven. It is not by chance that the Eastern Anaphoras and the Latin Eucharistic Prayers honour Mary, the ever-Virgin Mother of Jesus Christ our Lord and God, the angels, the holy apostles, the glorious martyrs and all the saints. This is an aspect of the Eucharist which merits greater attention: in celebrating the sacrifice of the Lamb, we are united to the heavenly “liturgy” and become part of that great multitude which cries out: “Salvation belongs to our God who sits upon the throne, and to the Lamb!” (Rev 7:10). [...]
20. A significant consequence of the eschatological tension inherent in the Eucharist is also the fact that it spurs us on our journey through history and plants a seed of living hope in our daily commitment to the work before us. Certainly the Christian vision leads to the expectation of “new heavens” and “a new earth” (Rev 21:1), but this increases, rather than lessens, our sense of responsibility for the world today.
[...] Lastly, the Magnificat reflects the eschatological tension of the Eucharist. Every time the Son of God comes again to us in the “poverty” of the sacramental signs of bread and wine, the seeds of that new history wherein the mighty are “put down from their thrones” and “those of low degree are exalted” (cf. Lk 1:52), take root in the world. Mary sings of the “new heavens” and the “new earth” which find in the Eucharist their anticipation and in some sense their programme and plan.